Symmetric convolution

In mathematics, symmetric convolution is a special subset of convolution operations in which the convolution kernel is symmetric across its zero point. Many common convolution-based processes such as Gaussian blur and taking the derivative of a signal in frequency-space are symmetric and this property can be exploited to make these operations easier to evaluate.

Convolution theorem

The convolution theorem states that a convolution in the real domain can be represented as a pointwise multiplication across the frequency domain of a Fourier transform. Since sine and cosine transforms are related transforms a modified version of the convolution theorem can be applied, in which the circular convolution is replaced with the concept of symmetric convolution. When using these transforms to compute discrete symmetric convolutions, caveats exist as discrete sine transforms (DSTs) and discrete cosine transforms (DCTs) can be counter-intuitively incompatible for computing symmetric convolution, i.e. symmetric convolution can only be computed between a fixed set of compatible transforms.

Mutually compatible transforms

In order to compute symmetric convolution effectively, one must know which possible frequency domains the inputs and outputs to the convolution can be and tailor the symmetries of the transforms to the required symmetries of the convolution.

The following table documents which combinations of the domains from the main eight commonly used DST I-IV and DCT I-IV satisfy $f*g=h$ where $*$ represents the symmetric convolution operator.

f g h DCT-I DCT-I DCT-I DCT-I DST-I DST-I DST-I DST-I -DCT-I DCT-II DCT-I DCT-II DCT-II DST-I DST-II DST-II DCT-I DST-II DST-II DST-I -DCT-II DCT-II DCT-II DCT-I DCT-II DST-II DST-I DST-II DST-II -DCT-I

f g h DCT-III DCT-III DCT-III DCT-III DST-III DST-III DST-III DST-III -DCT-III DCT-IV DCT-III DCT-IV DCT-IV DST-III DST-IV DST-IV DCT-III DST-IV DST-IV DST-III -DCT-IV DCT-IV DCT-IV DCT-III DCT-IV DST-IV DST-III DST-IV DST-IV -DCT-III

Forward transforms of $f$, $g$ and $h$, through the transforms specified should allow the symmetric convolution to be computed as a pointwise multiplication with any excess frequencies set to zero. Possibilities for symmetric convolutions involving DSTs and DCTs V-VIII derived from the discrete Fourier transforms (DFTs) of odd logical order can be determined by adding four to each type in the above tables.

Advantages of specialised symmetric transforms

There are a number of advantages to computing symmetric convolutions in DSTs and DCTs in comparison with the more common circular convolution with the Fourier transform.

The boundary conditions implicit in DSTs and DCTs create edge effects that are often more in keeping with neighbouring data.

Implicit symmetry means that only data unable to be inferred by symmetry is required, in a signal only the positive half of the convolution kernel needs to be specified, the domain will implicitly construct the other half of the data. On a transform of $N$ points of data, the convolution kernel can have $2N$ points of real domain data where the other $N$ consists of symmetric data.

References

• S. A. Martucci, "Symmetric convolution and the discrete sine and cosine transforms," IEEE Trans. Sig. Processing SP-42, 1038-1051 (1994).